418 ON THE ATTRACTION OF AN INDEFINITELY THIN SHELL. [53 



Let E be an ellipsoid whose principal semi-axes are a, b, c; and let 

 E l be a confocal ellipsoid whose principal semi-axes are b lt c,. 



Then <*'-&' = <-&,*; &c. 



or a l *-a t = b l '-V = c 1 t -c'. 



First Solution. 



Let a, b, c be the semi-axes of E the interior surface of the attracting 

 shell, and let 1 + 1 be the ratio of similitude between the inner and outer 

 surfaces. 



Let M 1 (whose coordinates are x v y v z,) be the attracted point, a,, b,, c t 

 the semi-axes of a confocal ellipsoid through M 1} then 



a b c 



a*" \ y " c^ 



will be the coordinates of a point (M 1 suppose) on the ellipsoid E. 

 The equations to the normal to the ellipsoid E l at M 1 are 



i T7" 7 <> "\ ^ *> f~7 



a-X , b-l c'Z 



or a, 2 -- - = b{ - - - = Cj* - J - . 



! 2/1 1 



Take JT, y, Z the coordinates of a point M on this normal such that 



> 



a, 2 6 : - c, 1 



we see that the relation of M to ' M' is such that If is a corresponding 

 point to M' in the system of points whose relation is 



a b c 



Jrf is the point in which the normal to the external ellipsoid at M 1 

 meets the plane of contact of the cone of which M^ is the vertex and 

 which envelopes the attracting shell E. 



Let the attracting shell be divided into pairs of elements by means 

 of double cones of indefinitely small solid angle having their vertices at 

 the point M. 



Let one of these cones of solid angle 8<u intercept a pair of elements 

 of the shell E at P and Q. 



